Integral curvature estimates and geodesic limits for stable abelian Higgs fields

We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.